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Theorem fvreseq 5562
Description: Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
fvreseq  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
Distinct variable groups:    x, B    x, F    x, G
Allowed substitution hint:    A( x)

Proof of Theorem fvreseq
StepHypRef Expression
1 fnssres 5295 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
2 fnssres 5295 . . . 4  |-  ( ( G  Fn  A  /\  B  C_  A )  -> 
( G  |`  B )  Fn  B )
31, 2anim12i 551 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  ( G  Fn  A  /\  B  C_  A
) )  ->  (
( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B ) )
43anandirs 807 . 2  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B
) )
5 eqfnfv 5556 . . 3  |-  ( ( ( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B )  ->  (
( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( ( F  |`  B ) `  x )  =  ( ( G  |`  B ) `
 x ) ) )
6 fvres 5475 . . . . 5  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
7 fvres 5475 . . . . 5  |-  ( x  e.  B  ->  (
( G  |`  B ) `
 x )  =  ( G `  x
) )
86, 7eqeq12d 2272 . . . 4  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  =  ( ( G  |`  B ) `  x )  <->  ( F `  x )  =  ( G `  x ) ) )
98ralbiia 2550 . . 3  |-  ( A. x  e.  B  (
( F  |`  B ) `
 x )  =  ( ( G  |`  B ) `  x
)  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) )
105, 9syl6bb 254 . 2  |-  ( ( ( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B )  ->  (
( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
114, 10syl 17 1  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518    C_ wss 3127    |` cres 4663    Fn wfn 4668   ` cfv 4673
This theorem is referenced by:  tfrlem1  6359  tfr3  6383  fseqenlem1  7619  dchrresb  20461  rdgprc  23521  predreseq  23549  wfr3g  23625  frr3g  23650  bnj1536  28018  bnj1253  28179  bnj1280  28182
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-fv 4689
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